Devices for Forecasting Ratios in Hierarchies

ABSTRACT

Systems and methods for forecasting ratios in hierarchies are provided. Hierarchies can be formed that have components, including a numerator time series with values from input data, a denominator time series with values from input data, and a ratio time series of the numerator time series over the denominator time series. The components can be modeled to generate forecasted hierarchies. The forecasted hierarchies can be reconciled so that the forecasted hierarchies are statistically consistent throughout nodes of the forecasted hierarchies.

TECHNICAL FIELD

The present disclosure generally relates to computer-implemented,processor device-based systems and methods for forecasting ratios formedby a numerator time series and a denominator time series that aggregateto form hierarchies of time series.

BACKGROUND

Organizational decision-making processes can involve forecasting a timeseries. Systems and methods for forecasting ratio time series andforming hierarchical time series are desirable.

SUMMARY

In accordance with the teachings provided herein, systems and methodsfor forecasting ratio time series and forming hierarchical time seriesare provided.

For example, a computer-implemented method can include forming, on acomputing device, hierarchies having components that include a numeratortime series with values from input data, a denominator time series withvalues from the input data, and a ratio time series of the numeratortime series over the denominator time series. The components are modeledwith the computing device to generate forecasted hierarchies. Theforecasted hierarchies are reconciled so that the forecasted hierarchiesare statistically consistent throughout nodes of the forecastedhierarchies.

In another example, a system is provided that includes a server device.The server device includes a processor and a non-transitorycomputer-readable storage medium containing instructions that whenexecuted on the processor cause the processor to perform operations. Theoperations include forming hierarchies having components that include anumerator time series with values from input data, a denominator timeseries with values from the input data, and a ratio time series of thenumerator time series over the denominator time series. The operationsalso include modeling the components to generate forecasted hierarchiesand reconciling the forecasted hierarchies so that the forecastedhierarchies are statistically consistent throughout nodes of theforecasted hierarchies.

In another example, a computer-program product tangibly embodied in anon-transitory machine-readable storage medium is provided that includesinstructions that can cause a data processing apparatus to formhierarchies having components that include a numerator time series withvalues from input data, a denominator time series with values from theinput data, and a ratio time series of the numerator time series overthe denominator time series. The components are modeled to generateforecasted hierarchies. The forecasted hierarchies are reconciled sothat the forecasted hierarchies are statistically consistent throughoutnodes of the forecasted hierarchies.

The details of one or more embodiments are set forth in the accompanyingdrawings and the description below. Other features and aspects willbecome apparent from the description, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example of an environment that includes a dataprocessing device that can communicate with other devices using anetwork.

FIG. 2 shows an example of a data flow diagram that includes processesfor forecasting ratios.

FIG. 3 shows an example of a data flow diagram that includes processesfor forecasting components of hierarchies.

FIG. 4 shows an example of a data flow diagram that includes modelforecasting using a direct process.

FIG. 5 shows an example of a data flow diagram that includes modelforecasting using an indirect process.

FIG. 6 shows an example of a data flow diagram that includes modelforecasting using an adjusted process.

FIG. 7 shows an example of a data flow diagram that includesreconciliation using a direct process.

FIG. 8 shows an example of a data flow diagram that includesreconciliation using an indirect process.

FIG. 9 shows an example of a data flow diagram that includesreconciliation using an adjusted process.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

Certain aspects include systems and methods for forecasting ratiosformed by numerator time series and denominator time series that canaggregate to form hierarchical time series. Server device-based timeseries model selection techniques can be used to forecast ratios of timeseries organized in a hierarchical fashion. The forecasted ratios can bereconciled to achieve consistent reconciliation of the forecast ratiosthroughout the hierarchy. A hierarchy of these forecasts can be based oneither a pre-determined natural stratification or a business-drivensegmentation.

In some aspects, hierarchies can be formed that have componentsincluding a numerator time series with values from input data, adenominator time series with values from input data, and a ratio timeseries of the numerator time series over the denominator time series.The components at each node can be modeled and then reconciled so thatthe forecasted hierarchies are statistically consistent throughout nodesof the hierarchies.

Organizations can use forecasted ratios in a hierarchical context indecision-making processes. The historical time series related tonumerator and denominator can be aggregated with other related timeseries to form hierarchical time series. The ratio time series can alsoform a hierarchical time series. A hierarchy of these forecasts can bebased on either a pre-determined natural stratification or abusiness-driven segmentation.

For example, given the claim amount (i.e., severity) and the number ofclaims (i.e., counts) associated with loss and average loss forinsurance loss as computed by the ratio of severity and counts,forecasted ratios can provide forecasts for the average loss (i.e.,severity over counts). Claims can be organized by policies, geographicregion, demographics, etc.

Another example is insurance costs. Given the total costs related to aninsurance policy (i.e., costs) and the number of policy holders (i.e.,counts), forecasted ratios can provide forecasts of the cost per memberper month. Costs can be organized by policies, geographic region,demographics, etc.

Another example is manufacturing warranties. Given the total claims(i.e., failures) and the number of units sold (i.e., counts) formanufacturing warranty claims, forecasted ratios can provide forecastsof the incremental failure rate (i.e., failures over counts). Failurescan be organized by product line, type of warranty, factory, etc.

Another example is in a banking context. Forecasted ratios can providethe average balance given the total credit card balances (i.e.,exposure) and the number of cardholders (i.e., counts). Balances forbanking purposes can be organized by credit score, type of credit card,or another attribute.

FIG. 1 is an example of an environment in which certain aspects may beimplemented using a data processing device 100. The data processingdevice 100 includes a processor 102 that can execute code stored on atangible computer-readable medium in a memory 104 to cause the dataprocessing device 100 to perform actions. The data processing device 100may be any device that can process data and execute code that is a setof instructions to perform actions. Examples of the data processingdevice 100 include a database server, a web server, desktop personalcomputer, a laptop personal computer, a server device, a handheldcomputing device, and a mobile device.

Examples of the processor 102 include a microprocessor, anapplication-specific integrated circuit (ASIC), a state machine, orother suitable processor. The processor 102 may include one processor orany number of processors. The processor 102 can access code stored inthe memory 104 via a bus 106. The memory 104 may be any non-transitorycomputer-readable medium configured for tangibly embodying code and caninclude electronic, magnetic, or optical devices. Examples of the memory104 include random access memory (RAM), read-only memory (ROM), a floppydisk, compact disc, digital video device, magnetic disk, an ASIC, aconfigured processor, or other storage device.

Instructions can be stored in the memory 104 as executable code. Theinstructions can include processor-specific instructions generated by acompiler and/or an interpreter from code written in any suitablecomputer-programming language. The instructions can include anapplication, such as ratio forecasting engine 108, that, when executedby the processor 102, can cause the data processing device 100 to formhierarchies including ratio time series, forecast the hierarchies, andreconcile the hierarchies. Memory 104 includes a datastore 110 in whichthe forecasted hierarchies and reconciled hierarchies can be stored.

The data processing device 100 includes an input/output (I/O) interface112 through which the data processing device 100 can communicate withdevices and systems external to the data processing device 100. Forexample, the data processing device 100 can receive input data throughI/O interface 112 from sources, such as an input data storage device 114connected to the data processing device 100 or an input data storagedevice 116 through a network 118 connected to the data processing device100. Input data may be any type of data usable for forming a numeratortime series and/or a denominator time series. Network 118 may be anysuitable network. Examples of network 118 include a wide area network(WAN), a local area network (LAN), a wireless local area network (WLAN),and a combination of these and/or other types of networks.

FIG. 2 is a data flow diagram that depicts an example of certainprocesses that can be performed by the data processing device 100 ofFIG. 1.

As shown in FIG. 2, the data processing device 100 uses input data 202in a process 204 of forming hierarchies. Input data 202 can include datasets, such as data sets that include time series for certain data types,or otherwise data that the data processing device 100 organizes intotime series. The hierarchies include components 206, such as denominatortime series, numerator time series, and ratio time series formed by thedata processing device 100 executing the ratio forecasting engine 108.In some aspects, the numerator time series is aggregated, thedenominator time series is aggregated, and an aggregated ratio timeseries is generated in forming the hierarchies.

The data processing device 100 performs a process 208 of modeling thecomponents 206 of the hierarchies to generate forecasted hierarchies210. The components 206 can be modeled independent or jointly, such asmodeling the numerator time series and the denominator time seriesjointly at a node of the hierarchies. The model for a node can includepredictor variables that may be different for the numerator series, thedenominator series or the ratio series, or may differ across nodes. Thecomponents 206 can be modeled using various methodologies, including adirect process, an indirect process, and an adjusted process, applied toindividual or aggregated time series. In a direct process, a ratio modelprediction can be used to predict a ratio time series forecast directly.In an indirect process, the ratio time series forecast can be determinedbased on a numerator time series forecast and a denominator time seriesforecast. In some aspects, confidence bands are derived for theforecasted hierarchies using a transformation or simulation in theindirect process. In an adjusted process, the numerator time series canbe pre-adjusted by dividing the numerator time series by the denominatortime series appended with denominator time series future predictions,the pre-adjusted numerator time series can be modeled to generateadjusted predictions, and the adjusted predictions can be post-adjustedby multiplying the adjusted predictions by the denominator time seriesappended with denominator time series future predictions.

The data processing device 100 performs a process 212 of reconciliationprocessing on the forecasted hierarchies 210 to provide statisticallyconsistent forecasted hierarchies 214 that may be used fordecision-making purposes. Reconciliation processing can includereconciling the probability distribution, including the mean of theprediction distribution, the prediction standard errors or variances,and upper and lower confidence bands. Forecasted hierarchies 210 can bereconciled using various methodologies, including a directreconciliation process, an indirect reconciliation process, and anadjusted reconciliation process. A direct reconciliation process caninclude using a weight associated with each of the components inreconciling the forecasted hierarchies. The weight may be a time-varyingweight and may be determined using reconciled forecasts for thedenominator time series and reconciled forecasts for aggregatedenominator time series. An indirect reconciliation process can includedetermining reconciled ratio forecasts using reconciled forecasts forthe numerator time series and reconciled forecasts for the denominatortime series. An adjusted reconciliation process can include determiningreconciled ratio forecasts using reconciled forecasts for the numeratortime series appended to actual values and reconciled forecasts for thedenominator time series appended to actual values.

In some aspects, an override input 216 can be received by the dataprocessing device 100. The override input 216 may indicate a changethrough a hierarchy, such as a change to a numerator time series, adenominator time series, and/or a ratio time series. For example,business personnel may apply judgmental inputs for information notcaptured in historical input data. In response to the override input216, the data processing device 100 can model the components, includingthe change, to generate re-forecasted hierarchies, and can re-reconcilethe re-forecasted hierarchies to generate statistically consistentforecasted hierarchies 214 including the change.

FIG. 3 is a data flow diagram depicting an example of forecastingcomponents of hierarchies.

An input data set 300 can be used to form a numerator time series 302and a denominator time series 304. For example, numerator time seriesvalues can be represented by y_(i,t), where tε{t_(i) ^(b), . . . , t_(i)^(e)} is the time index for the i^(th) dependent series and where i=1, .. . , N. The aggregate of the individual numerator time series can berepresented by

$Y_{t} = {\sum\limits_{i = 1}^{N}\; {y_{i,t}.}}$

In the aggregation, missing values can be ignored. Denominator timeseries values can be represented by x_(i,t), where tε{t_(i) ^(b), . . ., t_(i) ^(e)} is the time index for the i^(th) dependent series andwhere i=1, . . . , N. The aggregate of the individual denominator timeseries can be represented by

$X_{t} = {\sum\limits_{i = 1}^{N}\; {x_{i,t}.}}$

In the aggregation, missing values can be ignored.

N can represent the number of series recorded in the time series dataset and i=1, . . . , N can represent the series index. T can representthe length of the series and t=1, . . . , T can represent the timeindex. The time index can be an ordered set of contiguous integersrepresenting time periods associated with equally spaced time intervals.In some aspects, the beginning and/or ending time index coincide. Inother aspects, the beginning and/or ending time index do not coincide.The relationship, tε{t_(i) ^(b), (t_(i) ^(b)), . . . , (t_(i) ^(e)),t_(i) ^(e)} can represent the time index, where t_(i) ^(b) and t_(i)^(e) represent the beginning and ending time index for the i^(th)series, respectively.

The ratio time series 306 can be computed 308 using the numerator timeseries 302 and the denominator time series 304. For example, ratio timeseries values may be the numerator time series values over thedenominator time series values and can be represented byz_(i,t)=y_(i,t)/x_(i,t), where tε{t_(i) ^(b), . . . , t_(i) ^(e)} is thetime index for the i^(th) dependent series, and i=1, . . . , N. Whenx_(i,t)=0, then z_(i,t) can be set to missing. Z_(t)=Y_(t)/X_(t)represents the ratio of aggregates of the numerator time series and thedenominator time series. Although possible,

${Z_{t} \neq {\sum\limits_{i = 1}^{N}\; z_{i,t}}},$

typically. That is, the ratio time series may not be able to beaggregated in the usual way. The ratio time series can represent averagevalues or percentages.

The numerator time series, the denominator time series, and the ratiotime series can have individual (i.e., disaggregate) and aggregateforms. Although processes applied to a two-level hierarchy aredescribed, multilevel hierarchies can be formed by continuing toaggregate the aggregates and processes can be applied to multilevelhierarchies.

Each of the numerator time series 302, denominator time series 304, andratio time series 306 can be applied to automatic forecasting processes310, 312, 314 to generate numerator forecasts 316, denominator forecasts318, and ratio forecasts 320, respectively. The automatic forecastingprocesses 310, 312, 314 can involve modeling using one or moretechniques, such as direct, indirect, and/or adjusted processes. Thenumerator time series 302 and denominator time series 304 may be modeledand forecast independently or jointly using vector time seriesforecasting techniques. The models can include predictor variables. Insome aspects, an extend time series process 322 can be applied to thedenominator time series 304 to generate an extended denominator series324 that can be used for modeling or reconciliation. A time series modelcan be used to forecast each component series. {circumflex over(x)}_(i,t), ŷ_(i,t), and {circumflex over (z)}_(i,t) represent the modelforecasts for the denominator time series 304, numerator time series302, and ratio time series 306, respectively. Typically, {circumflexover (z)}_(i,t)≠ŷ_(i,t)/{circumflex over (z)}_(i,t) except on occasion.{hacek over (x)}_(i,t), {hacek over (y)}_(i,t), {hacek over (z)}_(i,t)represent the denominator time series 304, numerator time series 302,and ratio time series 306, respectively, appended with futurepredictions (i.e., extended series). Typically. {hacek over(z)}_(i,t)≠{hacek over (y)}_(i,t)/{hacek over (z)}_(i,t) except onoccasion.

FIG. 4 is a data flow diagram depicting an example of model forecastingusing a direct process. Included in FIG. 4 are the numerator time series302, the numerator forecasts 316, the ratio time series 306, the ratioforecasts 320, denominator time series 304, denominator forecasts 318,and extended denominator series 324. The forecasts, which may be modelpredictions, can be used directly to predict the forecasts. For example,the numerator forecasts 316 can be considered direct numerator forecasts402, the ratio forecasts 320 can be considered direct ratio forecasts404, and the denominator forecasts 318 can be considered directdenominator forecasts 406. The direct ratio forecasts 404, which may beindividual or aggregate, can be determined without considering thenumerator time series 302 or the denominator time series 304. Confidencebands can be provided directly.

FIG. 5 is a data flow diagram depicting an example of model forecastingusing an indirect process. Included in FIG. 5 are the numerator timeseries 302, the numerator forecasts 316, the ratio time series 306, thedenominator time series 304, the denominator forecasts 318, and theextended denominator series 324. In an indirect process, the ratioforecast can be modeled based on the numerator forecasts 316 anddenominator forecasts 318, rather than directly. The numerator forecasts316 and the denominator forecasts 318 can be used in indirect ratioforecasts process 502 to produce indirect ratio forecasts 504. In someaspects, the indirect ratio forecasts 504 can be determined by thenumerator forecasts 316 over the denominator forecasts 318. For example,

${\hat{z}}_{i,t}^{I} = \frac{{\hat{y}}_{i,t}}{{\hat{x}}_{i,t}}$

can represent the indirect ratio forecasts 504. The indirect numeratorforecasts 506 can be determined directly from the numerator forecasts316 and the indirect denominator forecasts 508 can be determineddirectly from the denominator forecasts 318. The indirect ratioforecasts 504, the indirect numerator forecasts 506, and the indirectdenominator forecasts 508 may be individual or aggregate.

Confidence bands for the indirect process can be determined usingGeary-Hinkley transformation or Monte Carlo or other type ofsimulations. For example,

and

can indicate the prediction standard errors of

and {circumflex over (x)}_(i,t), respectively, and

can represent the contemporaneous correlation of ŷ_(i,t) and {circumflexover (x)}_(i,t), assuming that the distribution of ŷ_(i,t) and{circumflex over (x)}_(i,t) is Gaussian.

and

can indicate, respectively, the lower and upper confidence limits oflevel α for {circumflex over (z)}_(i,t). If {circumflex over (x)}_(i,t)is unlikely to be negative, the Geary-Hinkley transformation can be usedto find the approximate prediction bands for {circumflex over(z)}_(i,t). For example:

$s_{i,t} = {{g_{i,t}(z)} = \frac{{{\hat{x}}_{i,t}z} - {\hat{y}}_{i,t}}{\sqrt{{\text{?}z^{2}} - {2\text{?}} + \text{?}}}}$?indicates text missing or illegible when filed

The variable s_(i,t) can be approximately distributed according to astandard normal distribution. I_(αN) and u_(αN) can be, respectively,the lower and upper confidence limits of level α for a standard normaldistribution. Then,

l̂_(i, t, α) ≈ g_(i, t)⁻¹(l_(α)^(N))û_(i, t, α)^(z) ≈ g_(i, t)⁻¹(u_(α)^(N))

Monte Carlo simulations can be used to find empirical confidence bandsfor {circumflex over (z)}_(i,t). For each fixed time t and series indexi, an arbitrarily large value M can be selected and M sets

$\left\lbrack {{\overset{\sim}{x}}_{i,t,j}\mspace{14mu} {\overset{\sim}{y}}_{i,t,j}} \right\rbrack,{j = 1},\ldots \mspace{14mu},M$

can be drawn from a multivariate normal distribution with mean[{circumflex over (x)}_(i,t), ŷ_(i,t)] and covariance matrix

${\hat{\Sigma}}_{it} = \begin{bmatrix}\text{?} & \text{?} \\\text{?} & \text{?}\end{bmatrix}$ ?indicates text missing or illegible when filed

The following relationship can be defined:

${\overset{\sim}{z}}_{i,t,j}:=\frac{\text{?}}{\text{?}}$?indicates text missing or illegible when filed

j=1, . . . , M. Î_(i,t,α) ^(z) and û_(i,t,α) ^(z) can be taken to be,respectively, the α and (1−α) empirical quantiles of {circumflex over(z)}_(i,t,j), j=1, . . . , M.

FIG. 6 is a data flow diagram depicting an example of model forecastingusing an adjusted process. Included in FIG. 6 are the numerator timeseries 302, the numerator forecasts 316, the ratio time series 306, thedenominator time series 304, the denominator forecasts 318, and theextended denominator series 324. In an adjusted process, the numeratortime series 302 can be pre-adjusted in block 602 by the extendeddenominator series 324 to produce an adjusted numerator series 604. Thecomputation in block 602 can be represented by y_(i,t)^(A)=y_(i,t)/{tilde over (x)}_(i,t). An automatic forecasting process606 can be applied to the adjusted numerator series 604 to produce anadjusted numerator forecast 608, which may be adjusted predictionsrepresented by {tilde over (y)}_(i,t) ^(A) and can be consideredadjusted method ratio forecast 610. The adjusted numerator forecast 608can be post-adjusted in a re-adjusted numerator forecasts process 612using the extended denominator series 324 according to the followingrelationship, {hacek over (y)}_(i,t) ^(A)=ŷ_(i,t) ^(A){hacek over(x)}_(i,t) ^(A) to produce adjusted method numerator forecasts 614. Anadjusted method denominator forecast 616 can be determined directly fromdenominator forecasts 318.

Direct, indirect, and adjusted processes may be used with forecast modelcombination (i.e., ensemble) techniques.

FIG. 7 is a data flow diagram depicting an example of reconciliationusing a direct process. Included in FIG. 7 are direct numeratorforecasts (aggregate) 702, direct numerator forecasts (disaggregate)704, direct ratio forecasts (aggregate) 706, direct ratio forecasts(disaggregate) 708, direct denominator forecasts (aggregate) 710, anddirect denominator forecasts (disaggregate) 712. Although both aggregateand disaggregate versions of component forecasts are shown, the directprocess for reconciliation can be applied to one, but not both, of theaggregate or disaggregate versions of each of the component forecasts.

A reconciliation process 714 can be applied to the direct numeratorforecasts (aggregate) 702 to produce a reconciled numerator forecast(aggregate) 716. The reconciliation process 714 can be applied to thedirect numerator forecasts (disaggregate) 704 to produce reconcilednumerator forecast (disaggregate) 718. A reconciliation process 720 canbe applied to the direct denominator forecasts (aggregate) 710 toproduce a reconciled denominator forecast (aggregate) 722. Thereconciliation process 720 can be applied to the direct denominatorforecasts (disaggregate) 712 to produce reconciled denominator forecast(disaggregate) 724.

Ratio forecasts can be reconciled in a weighting process 726 using formsof weighting and reconciled numerator and reconciled denominatorforecasts to generate reconciled ratio forecast (aggregate) 728 and/orreconciled ratio forecast (disaggregate) 730. w_(t) can represent theweight associated with each series. The weight can be fixed across time.Reconciliation of the ratio forecasts can be performed using one or moreof the following relationships:

${\hat{Z}}_{t}^{R} = {\sum\limits_{i = 1}^{N}\; {w_{i}{\hat{z}}_{i,t}}}$

(bottom-up) or

${\hat{Z}}_{t} = {\sum\limits_{i = 1}^{N}\; {w_{i}{\hat{z}}_{i,t}^{R}}}$

(top-down).

Time-varying weights can be used in the weighting process 726 in someaspects. w_(i,t) can represent the weight associated with each seriesand a particular time.

The weighting can be varying over time. Reconciliation of the ratios canbe performed using one or more of the following relationships:

${\hat{Z}}_{t}^{R} = {\sum\limits_{i = 1}^{N}\; {w_{i,t}{\hat{z}}_{i,t}}}$

(bottom-up) or

${\hat{Z}}_{t} = {\sum\limits_{i = 1}^{N}\; {w_{i,t}{\hat{z}}_{i,t}^{R}}}$

(top-down). The weights can be changing over time (e.g., if the serieslengths differ) and the future weights may also be needed foraggregation. The future weights can be predicted. {hacek over (x)}_(i,t)^(R) can represent the reconciled forecasts for the denominator timeseries appended to the actual values. {hacek over (x)}_(i,t) ^(R) canrepresent the reconciled forecasts for the aggregate denominator timeseries appended to the actual values. The weights can be determinedusing one or more of the following relationships:

${{Bottom}\text{-}{Up}\mspace{14mu} {Weights}\text{:}\mspace{14mu} \omega_{i,t}} = {{\overset{\Cup}{x}}_{i,t}/{\overset{\Cup}{X}}_{i,t}^{R}}$${{Top}\text{-}{Down}\mspace{14mu} {Weights}\text{:}\mspace{14mu} \omega_{i,t}} = {\left( {{\hat{Z}}_{t}{\overset{\Cup}{x}}_{i,t}^{R}} \right)/\left( {{\overset{\Cup}{X}}_{i,t}{\sum\limits_{i = 1}^{N}\; {\hat{z}}_{i,t}}} \right)}$

Two forecasts may be needed to reconcile ratio forecasts at higherlevels of aggregation.

FIG. 8 is a data flow diagram depicting an example of reconciliationusing an indirect process. Included in FIG. 8 are indirect numeratorforecasts (aggregate) 802, indirect numerator forecasts (disaggregate)804, indirect ratio forecasts (aggregate) 806, indirect ratio forecasts(disaggregate) 808, indirect denominator forecasts (aggregate) 810, andindirect denominator forecasts (disaggregate) 812. Although bothaggregate and disaggregate versions of component forecasts are shown,the indirect process for reconciliation can be applied to one, but notboth, of the aggregate or disaggregate versions of each of the componentforecasts.

A reconciliation process 814 can be applied to the indirect numeratorforecasts (aggregate) 802 to produce a reconciled numerator forecast(aggregate) 816. The reconciliation process 814 can be applied to theindirect numerator forecasts (disaggregate) 804 to produce reconcilednumerator forecast (disaggregate) 818. A reconciliation process 820 canbe applied to the indirect denominator forecasts (aggregate) 810 toproduce a reconciled denominator forecast (aggregate) 822. Thereconciliation process 820 can be applied to the indirect denominatorforecasts (disaggregate) 812 to produce reconciled denominator forecast(disaggregate) 824.

The reconciled numerator forecast (aggregate) 816 and the reconcileddenominator forecast (aggregate) 822 can be used in a ratio computingprocess 826 to determine the reconciled ratio forecast (aggregate) 828.The reconciled numerator forecast (disaggregate) 818 and the reconcileddenominator forecast (disaggregate) 824 can be used in a ratio computingprocess 830 to determine the reconciled ratio forecast (disaggregate)832. For example, ŷ_(i,t) ^(R) and {circumflex over (x)}_(i,t) ^(R) canrepresent the reconciled forecasts for the numerator and denominatortime series. Ŷ_(i) ^(R) and {circumflex over (X)}_(i) ^(R) can representthe reconciled forecasts for the aggregate numerator and aggregatedenominator time series. The indirect reconciled ratio forecasts can bedetermined as follows: {circumflex over (z)}_(i,t) ^(IR)=ŷ_(i,t)^(R)/{circumflex over (x)}_(i,t) ^(R) and {circumflex over (Z)}_(t)^(IR)=Ŷ_(t) ^(R)/{circumflex over (X)}_(t) ^(R).

FIG. 9 is a data flow diagram depicting an example of reconciliationusing an adjusted process. Included in FIG. 9 are adjusted numeratorforecasts (aggregate) 902, adjusted numerator forecasts (disaggregate)904, adjusted ratio forecasts (aggregate) 906, adjusted ratio forecasts(disaggregate) 908, adjusted denominator forecasts (aggregate) 910, andadjusted denominator forecasts (disaggregate) 912. Although bothaggregate and disaggregate versions of component forecasts are shown,the adjusted process for reconciliation can be applied to one, but notboth, of the aggregate or disaggregate versions of each of the componentforecasts.

A reconciliation process 914 can be applied to the adjusted numeratorforecasts (aggregate) 902 to produce a reconciled numerator forecast(aggregate) 916. The reconciliation process 914 can be applied to theadjusted numerator forecasts (disaggregate) 904 to produce reconcilednumerator forecast (disaggregate) 918. A reconciliation process 920 canbe applied to the adjusted denominator forecasts (aggregate) 910 toproduce a reconciled denominator forecast (aggregate) 922. Thereconciliation process 920 can be applied to the adjusted denominatorforecasts (disaggregate) 912 to produce reconciled denominator forecast(disaggregate) 924.

The reconciled numerator forecast (aggregate) 916 and the reconcileddenominator forecast (aggregate) 922 can be used in a ratio computingprocess 926 to determine the reconciled ratio forecast (aggregate) 928.The reconciled numerator forecast (disaggregate) 918 and the reconcileddenominator forecast (disaggregate) 924 can be used in a ratio computingprocess 930 to determine the reconciled ratio forecast (disaggregate)932. For example, {hacek over (x)}_(i,t) ^(R) can represent thereconciled forecasts for the denominator time series appended to theactual values. {hacek over (X)}_(t) ^(R) can represent the reconciledforecasts for the aggregate denominator time series appended to theactual values. {hacek over (y)}_(i,t) ^(AR) can represent the reconciledforecasts for the adjusted numerator series. {hacek over (Y)}_(i,t)^(AR) can represent the reconciled forecasts for the adjusted aggregatenumerator time series. The adjusted reconciled ratio forecasts can bedetermined using the following relationships: {circumflex over(z)}_(i,t) ^(AR)={hacek over (y)}_(i,t) ^(R)/{hacek over (x)}_(i,t) ^(R)and {circumflex over (Z)}_(t) ^(AR)={hacek over (Y)}_(t) ^(AR)/{hacekover (X)}_(t) ^(R).

The following is an example of using certain aspects in forecastingratios in hierarchies. A large manufacturing company is seeking toforecast the amount of cash reserves that it needs to retain forcovering warranty claims on its products. This problem can involveaccounting processes that are less concerned with under-allocated versus“lazy” capital (i.e., respectively: money that is not available to coverclaims versus money that is tied up in reserve and not available forinvestment). The practical application of the problem can be focused ontwo areas: (1) accurate financial reporting where projected profits areimpacted by over/under allocation of expected reserves, and (2) customerperceptions about the stability and quality of the product line.

Using the screening process, the problem could be formulated inforecasting the total cost of claims over time directly, which mayinclude some adjustment for expected growth and inflation.Alternatively, the total claims or some form of a failure rate (e.g.,total claims per units in the field) can be forecasted. Forecasting as aratio can provide a form of consistency through analysis of theincremental failure rate (i.e., number of units sold that failed dividedby the number of units sold in the field). This can assume some degreeof stability based on classical reliability analysis that projects aparametric distribution curve based on a long-running and slowlyevolving statistical quality control process.

The components of the forecast can be known. For example, the numeratorcan be the number of actual failures and the denominator can be theembedded base for potential failures. The denominator may bedeterministic and/or controllable. For example, the company can have aplanned production cycle for an extended period of time, although marketdemand may cause adjustments. Market pressure, planned recalls,promotional warranty offerings, length of warranty, availability ofparts, and other causal factors may influence the numerator,denominator, or both. Low volatility may be expected on the denominator.Moderate volatility may be expected on the numerator that could varybased on the age of the units. The historical data streams can be robustfor the numerator and denominator. The additional internal/externalcovariates, however, may not be accurately and consistently capturedover a long period.

A hierarchical forecast can be used since the reserves may bepartitioned into a geographic segment (e.g., manufacturing plant) andproduct line (e.g., refrigerators, washer/dryers, HVAC, etc.). Themanufacturing can be assumed to be distributed equitably across plants.If the company is well-established, the customer base for the morepopular product lines may be somewhat similar. An exception may be“specialty” products that are only sold to specific markets. Plantdefinitions can be relatively stable unless a serious economicupturn/downturn warrants either expansion or closure. In that case, someof the historical plant data may need to be reclassified hierarchically(in the case of consolidation) or re-distributed (e.g., new plants).Product lines (at a macro level) may be relatively stable (although newmodels may be less so).

A direct process can be applied since, for example, the incrementalfailure rate (IFR) is an appropriate metric, the numerator variable(i.e., number of units in the field) is somewhat deterministic andstable, the appropriate hierarchy (e.g., top line, plant, product, etc.)is stable and equitably segmented. Furthermore, an additionalhierarchical level can be added that reflects the IFR for products of acertain age. For example, separate IFR's can be forecasted forone-month-old units, two-month-old, etc., with the end date based on thewarranty. The expanded hierarchy can include: (top line, plant, product,age). Bottom-up reconciliation can be used if the categories containsufficient data. Furthermore, a separate forecast for cost/claim can beperhaps, such as with an indirect process and involve a different levelof forecast segmentation (hierarchy) more focused on type of claim(i.e., much more erratic and volatile). In addition, the IFR forecastscan be matched with the projected volumes and cost/claim (adjusted forinflation), thus forming a picture of the requisite reserves.

Embodiments of the subject matter and the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, or in computer software, firmware, or hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. Embodiments ofthe subject matter described in this specification can be implemented asone or more computer program products, i.e., one or more modules ofcomputer program instructions encoded on a computer readable medium forexecution by, or to control the operation of, data processing apparatus.

The computer readable medium can be a machine readable storage device, amachine readable storage substrate, a memory device, a composition ofmatter effecting a machine readable propagated communication, or acombination of one or more of them. The term “data processing device”encompasses all apparatus, devices, and machines for processing data,including by way of example a programmable processor, a computer, ormultiple processors or computers. The device can include, in addition tohardware, code that creates an execution environment for the computerprogram in question, e.g., code that constitutes processor firmware, aprotocol stack, a database management system, an operating system, or acombination of one or more of them.

A computer program (also known as a program, software, softwareapplication, script, or code), can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a standalone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., on or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and a device can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC.

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Moreover, a computer can be embedded inanother device, e.g., a mobile telephone, a personal digital assistant(PDA), a mobile audio player, a Global Positioning System (GPS)receiver, to name just a few. Computer readable media suitable forstoring computer program instructions and data include all forms ofnonvolatile memory, media, and memory devices, including by way ofexample semiconductor memory devices, e.g., EPROM, EEPROM, and flashmemory devices; magnetic disks, e.g., internal hard disks or removabledisks; magneto optical disks; and CD ROM and DVD ROM disks. Theprocessor and the memory can be supplemented by, or incorporated in,special purpose logic circuitry.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube) to LCD (liquidcrystal display) monitor, for displaying information to the user and akeyboard and a pointing device, e.g., a mouse or a trackball, by whichthe user can provide input to the computer. Other kinds of devices canbe used to provide for interaction with a user as well; for example,feedback provided to the user can be any form of sensory feedback, e.g.,visual feedback, auditory feedback, or tactile feedback; and input fromthe user can be received in any from, including acoustic, speech, ortactile input.

Embodiments of the subject matter described in this specification can beimplemented in a computing system that includes a back end component,e.g., as a data server, or that includes a middleware component, e.g.,an application server, or that includes a front end component, e.g., aclient computer having a graphical user interface or a Web browserthrough which a user can interact with an implementation of the subjectmatter described in this specification, or any combination of one ormore such back end, middleware, or front end components. The componentsof the system can be interconnected by any form or medium of digitaldata communication, e.g., a communication network. Examples ofcommunication networks include a LAN and a WAN, e.g., the Internet.

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client server relationship to each other.

While this specification contains many specifics, these should not beconstrued as limitations on the scope or of what may be claimed, butrather as descriptions of features specific to particular embodiments.Certain features that are described in this specification in the contextor separate embodiments can also be implemented in combination in asingle embodiment. Conversely, various features that are described inthe context of a single embodiment can also be implemented in multipleembodiments separately or in any suitable subcombination. Moreover,although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the embodiments described above should not be understoodas requiring such separation in all embodiments, and it should beunderstood that the described program components and systems cangenerally be integrated together in a single software product orpackaged into multiple software products.

Thus, particular embodiments have been described. Other embodiments arewithin the scope of the following claims. For example, the actionsrecited in the claims can be performed in a different order and stillachieve desirable results.

What is claimed is:
 1. A computer-implemented method, comprising:forming, on a computing device, hierarchies having components thatinclude a numerator time series with values from input data, adenominator time series with values from the input data, and a ratiotime series of the numerator time series over the denominator timeseries; modeling, with the computing device, the components to generateforecasted hierarchies; and reconciling the forecasted hierarchies sothat the forecasted hierarchies are statistically consistent throughoutnodes of the forecasted hierarchies.
 2. The method of claim 1, furthercomprising: responsive to receiving an override that includes a changeto at least one of the hierarchies, modeling the components to generatere-forecasted hierarchies; and re-reconciling the re-forecastedhierarchies.
 3. The method of claim 1, wherein modeling the componentsincludes modeling each component independently from each other.
 4. Themethod of claim 1, wherein modeling the components includes modeling thenumerator time series and the denominator time series jointly at a nodeof the hierarchies.
 5. The method of claim 1, wherein modeling thecomponents includes using a ratio model prediction directly to predict aratio time series forecast.
 6. The method of claim 5, whereinreconciling the forecasted hierarchies includes using a weightassociated with each of the components.
 7. The method of claim 6,wherein the weight is a time-varying weight.
 8. The method of claim 7,further comprising: determining the weight using reconciled forecastsfor the denominator time series and reconciled forecasts for anaggregate denominator time series.
 9. The method of claim 1, whereinmodeling the components includes determining the ratio time seriesforecast based on a numerator time series forecast and a denominatortime series forecast.
 10. The method of claim 9, wherein reconciling theforecasted hierarchies includes determining reconciled ratio forecastsusing reconciled forecasts for the numerator time series and reconciledforecasts for the denominator time series.
 11. The method of claim 9,wherein modeling the components includes deriving confidence bands forthe forecasted hierarchies using a transformation or simulation.
 12. Themethod of claim 1, wherein modeling the components includes:pre-adjusting the numerator time series by dividing the numerator timeseries by the denominator time series appended with denominator timeseries future predictions; modeling the pre-adjusted numerator timeseries to generate adjusted predictions; and post-adjusting the adjustedpredictions by multiplying the adjusted predictions by the denominatortime series appended with denominator time series future predictions.13. The method of claim 12, wherein reconciling the forecastedhierarchies includes: determining reconciled ratio forecasts usingreconciled forecasts for the numerator time series appended to actualvalues and reconciled forecasts for the denominator time series appendedto actual values.
 14. The method of claim 1, wherein forming thehierarchies includes: aggregating the numerator time series; aggregatingthe denominator time series; and generating an aggregated ratio of theaggregated numerator time series and the aggregated denominator timeseries.
 15. The method of claim 14, wherein modeling the componentsincludes using ratio model prediction directly to predict a ratioforecast for the aggregated ratio.
 16. The method of claim 14, whereinmodeling the components includes predicting a ratio forecast for theaggregated ratio using a forecast of the aggregated numerator timeseries and a forecast of the aggregated denominator time series.
 17. Themethod of claim 14, wherein modeling the components includes:pre-adjusting the aggregated numerator time series by dividing theaggregated numerator time series by the aggregated denominator timeseries appended with future predictions; modeling the pre-adjustedaggregated numerator time series to generate adjusted aggregatedpredictions for the numerator time series; and post-adjusting theadjusted aggregated predictions by multiplying the adjusted aggregatedpredictions with the aggregated denominator time series appended withthe future predictions.
 18. A system, comprising: a server device thatincludes: a processor, and a non-transitory computer-readable storagemedium containing instructions that when executed on the processor causethe processor to perform operations including: forming hierarchieshaving components that include a numerator time series with values frominput data, a denominator time series with values from the input data,and a ratio time series of the numerator time series over thedenominator time series; modeling the components to generate forecastedhierarchies; and reconciling the forecasted hierarchies so that theforecasted hierarchies are statistically consistent throughout nodes ofthe forecasted hierarchies.
 19. The system of claim 18, wherein theserver device includes instructions configured to cause the processor toperform operations including: responsive to receiving an override thatincludes a change to at least one of the hierarchies, modeling thecomponents to generate re-forecasted hierarchies; and re-reconciling there-forecasted hierarchies.
 20. The system of claim 18, wherein modelingthe components includes modeling each component independently from eachother.
 21. The system of claim 18, wherein modeling the componentsincludes modeling the numerator time series and the denominator timeseries jointly at a node of the hierarchies.
 22. The system of claim 18,wherein modeling the components includes using a ratio model predictiondirectly to predict a ratio time series forecast.
 23. The system ofclaim 22, wherein the server device includes instructions configured tocause the processor to perform operations including: reconciling theforecasted hierarchies by using a weight associated with each of thecomponents.
 24. The system of claim 23, wherein the weight is atime-varying weight.
 25. The system of claim 24, wherein the serverdevice includes instructions configured to cause the processor toperform operations including: determining the weight using reconciledforecasts for the denominator time series and reconciled forecasts foraggregate denominator time series.
 26. The system of claim 18, whereinmodeling the components includes determining the ratio time seriesforecast based on a numerator time series forecast and a denominatortime series forecast.
 27. The system of claim 26, wherein reconcilingthe forecasted hierarchies includes determining reconciled ratioforecasts using reconciled forecasts for the numerator time series andreconciled forecasts for the denominator time series.
 28. The system ofclaim 26, wherein modeling the components includes deriving confidencebands for the forecasted hierarchies using a transformation orsimulation.
 29. The system of claim 18, wherein the server deviceincludes instructions configured to cause the processor to performoperations using the modeling the components including: pre-adjustingthe numerator time series by dividing the numerator time series by thedenominator time series appended with denominator time series futurepredictions; modeling the pre-adjusted numerator time series to generateadjusted predictions; and post-adjusting the adjusted predictions bymultiplying the adjusted predictions by the denominator time seriesappended with denominator time series future predictions.
 30. The systemof claim 29, wherein the server device includes instructions configuredto cause the processor to perform operations including: reconciling theforecasted hierarchies by determining reconciled ratio forecasts usingreconciled forecasts for the numerator time series appended to actualvalues and reconciled forecasts for the denominator time series appendedto actual values.
 31. The system of claim 18, wherein the server deviceincludes instructions configured to cause the processor to performoperations including forming the hierarchies by operations including:aggregating the numerator time series; aggregating the denominator timeseries; and generating an aggregated ratio of the aggregated numeratortime series and the aggregated denominator time series.
 32. The systemof claim 31, wherein modeling the components includes using ratio modelprediction directly to predict a ratio forecast for the aggregatedratio.
 33. The system of claim 31, wherein modeling the componentsincludes predicting a ratio forecast for the aggregated ratio using aforecast of the aggregated numerator time series and a forecast of theaggregated denominator time series.
 34. The system of claim 31, whereinthe server device includes instructions configured to cause theprocessor to perform operations including modeling the components withoperations including: pre-adjusting the aggregated numerator time seriesby dividing the aggregated numerator time series by the aggregateddenominator time series appended with future predictions; modeling thepre-adjusted aggregated numerator time series to generate adjustedaggregated predictions for the numerator time series; and post-adjustingthe adjusted aggregated predictions by multiplying the adjustedaggregated predictions with the aggregated denominator time seriesappended with the future predictions.
 35. A computer-program producttangibly embodied in a non-transitory machine-readable storage medium,including instructions configured to cause a data processing apparatusto: form hierarchies having components that include a numerator timeseries with values from input data, a denominator time series withvalues from the input data, and a ratio time series of the numeratortime series over the denominator time series; model the components togenerate forecasted hierarchies; and reconcile the forecastedhierarchies so that the forecasted hierarchies are statisticallyconsistent throughout nodes of the forecasted hierarchies.
 36. Thecomputer-program product of claim 35, further comprising instructionsconfigured to cause the data processing apparatus to: responsive toreceiving an override that includes a change to at least one of thehierarchies, model the components to generate re-forecasted hierarchies;and re-reconcile the re-forecasted hierarchies.
 37. The computer-programproduct of claim 35, wherein instructions configured to cause the dataprocessing apparatus to model the components includes instructions formodeling each component independently from each other.
 38. Thecomputer-program product of claim 35, wherein instructions configured tocause the data processing apparatus to model the components includesinstructions for modeling the numerator time series and the denominatortime series jointly at a node of the hierarchies.
 39. Thecomputer-program product of claim 35, wherein instructions configured tocause the data processing apparatus to model the components includesinstructions for using a ratio model prediction directly to predict aratio time series forecast.
 40. The computer-program product of claim39, wherein instructions configured to cause the data processingapparatus to reconcile the forecasted hierarchies includes instructionsfor using a weight associated with each of the components.
 41. Thecomputer-program product of claim 40, wherein the weight is atime-varying weight.
 42. The computer-program product of claim 41,further comprising instructions configured to cause the data processingapparatus to determine the weight using reconciled forecasts for thedenominator time series and reconciled forecasts for aggregatedenominator time series.
 43. The computer-program product of claim 35,wherein instructions configured to cause the data processing apparatusto model the components includes instructions for determining the ratiotime series forecast based on a numerator time series forecast and adenominator time series forecast.
 44. The computer-program product ofclaim 43, wherein instructions configured to cause the data processingapparatus to reconcile the forecasted hierarchies includes instructionsfor determining reconciled ratio forecasts using reconciled forecastsfor the numerator time series and reconciled forecasts for thedenominator time series.
 45. The computer-program product of claim 43,wherein instructions configured to cause the data processing apparatusto model the components includes instructions for deriving confidencebands for the forecasted hierarchies using a transformation orsimulation.
 46. The computer-program product of claim 43, whereininstructions configured to cause the data processing apparatus to modelthe components includes instructions for: pre-adjusting the numeratortime series by dividing the numerator time series by the denominatortime series appended with denominator time series future predictions;modeling the pre-adjusted numerator time series to generate adjustedpredictions; and post-adjusting the adjusted predictions by multiplyingthe adjusted predictions by the denominator time series appended withdenominator time series future predictions.
 47. The computer-programproduct of claim 46, wherein instructions configured to cause the dataprocessing apparatus to reconcile the forecasted hierarchies includesinstructions for determining reconciled ratio forecasts using reconciledforecasts for the numerator time series appended to actual values andreconciled forecasts for the denominator time series appended to actualvalues.
 48. The computer-program product of claim 35, whereininstructions configured to cause the data processing apparatus to formthe hierarchies includes instructions for: aggregating the numeratortime series; aggregating the denominator time series; and generating anaggregated ratio of the aggregated numerator time series and theaggregated denominator time series.
 49. The computer-program product ofclaim 48, wherein instructions configured to cause the data processingapparatus to model the components includes instructions for using ratiomodel prediction directly to predict a ratio forecast for the aggregatedratio.
 50. The computer-program product of claim 49, whereininstructions configured to cause the data processing apparatus to modelthe components includes instructions for predicting a ratio forecast forthe aggregated ratio using a forecast of the aggregated numerator timeseries and a forecast of the aggregated denominator time series.
 51. Thecomputer-program product of claim 49, wherein instructions configured tocause the data processing apparatus to model the components includesinstructions for: pre-adjusting the aggregated numerator time series bydividing the aggregated numerator time series by the aggregateddenominator time series appended with future predictions; modeling thepre-adjusted aggregated numerator time series to generate adjustedaggregated predictions for the numerator time series; and post-adjustingthe adjusted aggregated predictions by multiplying the adjustedaggregated predictions with the aggregated denominator time seriesappended with the future predictions.